File Name: optical resonance and two level atoms .zip
Resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom. Typically the photon contained electromagnetic field is applied to the two-level atom through the use of a monochromatic laser.
A two-level atom is a specific type of two-state system in which the atom can be found in the two possible states. The two possible states are if an electron is found in its ground state or the excited state. In many experiments an atom of lithium is used because it can be closely modeled to a two-level atom as the excited states of the singular electron are separated by large enough energy gaps to significantly reduce the possibility of the electron jumping to a higher excited state.
Thus it allows for easier frequency tuning of the applied laser as frequencies further off resonance can be used while still driving the electron to jump to only the first excited state. Once the atom is excited, it will release a photon at the frequency of the absorbed photon within the range of the detuning of the laser from the natural resonance of the atom.
The mechanism for this release is the spontaneous decay of the atom. The emitted photon is released in an arbitrary direction. While the transition between two specific energy levels is the dominant mechanism in resonance fluorescence, experimentally other transitions will play a very small role and thus must be taken into account when analyzing results. The other transitions will lead to emission of a photon of a different atomic transition with much lower energy which will lead to "dark" periods of resonance fluorescence.
Thus the energy of the system is described entirely through an electric dipole interaction between the atom and field with the resulting hamiltonian being described by. Now that the dynamics of the field with respect to the states of the atom has been described, the mechanism through which photons are released from the atom as the electron falls from the excited state to the ground state, Spontaneous Emission , can be examined.
Spontaneous emission is when an excited electron arbitrarily decays to the ground state emitting a photon. As the electromagnetic field is coupled to the state of the atom, and the atom can only absorb a single photon before having to decay, the most basic case then is if the field only contains a single photon.
During this process the decay of the expectation values of the above operators follow the following relations. So the atom decays exponentially and the atomic dipole moment shall oscillate. The dipole moment oscillates due to the Lamb shift, which is a shift in the energy levels of the atom due to fluctuations of the field. It is imperative, however, to look at fluorescence in the presence of a field with many photons, as this is a much more general case.
This is the case in which the atom goes through many excitation cycles. This allows for the operators which comprise the field to act on the coherent state and thus be replaced with eigenvalues. Thus we can simplify the equations by allowing operators to be turned into constants.
The field can then be described much more classically than a quantized field normally would be able to. As a result, we are able to find the expectation value of the electric field for the retarded time. There are two general types of excitations produced by fields.
Thus the dynamics of a two-level atom can be accurately modeled by a photon in an interferometer. There are several limits that can be analyzed to make the study of resonance fluorescence easier. The first of these is the approximations associated with the Weak Field Limit , where the square modulus of the Rabi frequency of the field that is coupled to two-level atom is much smaller than the rate of spontaneous emission of the atom.
This means that the difference in the population between the excited state of the atom and the ground state of the atom is approximately independent of time. Thus it is clear that when an electric field is applied to the atom, the dipole of the atom oscillates according to driving frequency and not the natural frequency of the atom.
The result is that the two-level atom behaves exactly as a driven oscillator and continues scattering photons so long as the driving field remains coupled to the atom. The weak field approximation is also used in approaching two-time correlation functions.
These are the result of the Markovian processes of the quantum fluctuations of the system. The Strong Field Limit is the exact opposite limit to the weak field where the square modulus of the Rabi frequency of the electromagnetic field is much larger than the rate of spontaneous emission of the two-level atom.
When a strong field is applied to the atom, a single peak is no longer observed in fluorescent light's radiation spectrum. Instead, other peaks begin appearing on either side of the original peak. These are known as side bands. The sidebands are a result of the Rabi oscillations of the field causing a modulation in the dipole moment of the atom. This is known as dynamic Stark splitting and is the cause for the Mollow triplet, which is a characteristic energy spectrum found in Resonance fluorescence.
An interesting phenomena arises in the Mollow triplet where both of the sideband peaks have a width different than that of the central peak. Thus the spectrum would vanish in a steady state solution, which is not the actual case. The solution that does allow for a steady state solution must take the form of a two-time correlation function as opposed to the above one-time correlation function.
This solution appears as. The study of correlation functions is critical to the study of quantum optics as the Fourier transform of the correlation function is the energy spectral density. Thus the two-time correlation function is a useful tool in the calculation of the energy spectrum for a given system.
While correlation functions can more easily be described using limits of the strength of the field and limits placed on the time of the system, they can be found more generally as well. For resonance fluorescence, the most important correlation functions are. The correlation function associated with the spectral density of resonance fluorescence is reliant on the electric field.
Thus once the constant K has been determined, the result is equivalent to. Photon anti-bunching is the process in Resonance Fluorescence through which rate at which photons are emitted by a two-level atom is limited. A two-level atom is only capable of absorbing a photon from the driving electromagnetic field after a certain period of time has passed. As the atom cannot absorb a photon, it is unable to emit one and thus there is a restriction on the spectral density.
The physical idea behind photon anti-bunching is that while the atom itself is ready to be excited as soon as it releases its previous photon, the electromagnetic field created by the laser takes time to excite the atom. Double Resonance  is the phenomena when an additional magnetic field is applied to a two-level atom in addition to the typical electromagnetic field used to drive resonance fluorescence.
Thus resonance is achievable not only about the possible energy-levels of a two-level atom, but also about the sub-levels in the energy created by lifting the degeneracy of the level. If the applied magnetic field is tuned properly, the polarization of resonance fluorescence can be used to describe the composition of the excited state.
Any two state system can be modeled as a two-level atom. This leads to many systems being described as an "Artificial Atom". For instance a superconducting loop which can create a magnetic flux passing through it can act as an artificial atom as the current can induce a magnetic flux in either direction through the loop depending on whether the current is clockwise or counterclockwise. This models the dipole interaction of the atom with a 1-D electromagnetic wave. It is easy to see that this is truly analogous to a real two-level atom due to the fact that the fluorescence appears in the spectrum as the Mollow triplet, precisely like a true two-level atom.
These artificial atoms are often used to explore the phenomena of quantum coherence. This allows for the study of squeezed light which is known for creating more precise measurements. It is difficult to explore the resonance fluorescence of squeezed light in a typical two-level atom as all modes of the electromagnetic field must be squeezed which cannot easily be accomplished.
In an artificial atom, the number of possible modes of the field is significantly limited allowing for easier study of squeezed light. In D. Toyli et al. The implication of this study is it allows for resonance fluorescence to assist in qubit readout for squeezed light.
The qubit used in the study was an aluminum transmon circuit that was then coupled to a 3-D aluminum cavity. Extra silicon chips were introduced to the cavity to assist in the tuning of resonance to that of the cavity. The majority of the detuning that did occur was a result of the degeneration of the qubit over time. A quantum dot is a semiconductor nano-particle that is often used in quantum optical systems.
This includes their ability to be placed in optical microcavities where they can act as two-level systems. In this process, quantum dots are placed in cavities which allow for the discretization of the possible energy states of the quantum dot coupled with the vacuum field.
The vacuum field is then replaced by an excitation field and resonance fluorescence is observed. Current technology only allows for population of the dot in an excited state not necessarily always the same , and relaxation of the quantum dot back to its ground state.
Direct excitation followed by ground state collection was not achieved until recently. This is mainly due to the fact that as a result of the size of quantum dots, defects and contaminants create fluorescence of their own apart from the quantum dot.
This desired manipulation has been achieved by quantum dots by themselves through a number of techniques including four-wave mixing and differential reflectivity, however no techniques had shown it to occur in cavities until Resonance fluorescence has been seen in a single self-assembled quantum dot as presented by Muller among others in Thus the quantum dot was not placed in the cavity, but instead created in it.
They then coupled a strong in-plane polarized tunable continuous-wave laser to the quantum dot and were able to observe resonance fluorescence from the quantum dot. In addition to the excitation of the quantum dot that was achieved, they were also able to collect the photon that was emitted with a micro-PL setup. This allows for resonant coherent control of the ground state of the quantum dot while also collecting the photons emitted from the fluorescence. In , G. Wrigge, I. Gerhardt, J.
Hwang, G. Zumofen, and V. Sandoghdar Developed and efficient method to observe resonance fluorescence for an entire molecule as opposed to its typical observation in a single atom. They used a tunable dye laser to excite the dye molecules in their sample.
Due to the fact that they could only have one source at a time, the proportion of shot noise to actual data was much higher than normal. The sample which they excited was a Shpol'skii matrix which they had doped with the dyes they wished to use, dibenzanthanthrene. To improve the accuracy of the results, single-molecule fluorescence-excitation spectroscopy was used.
The actual process for measuring the resonance was measuring the interference between the laser beam and the photons that were scattered from the molecule. Thus the laser was passed over the sample, resulting in several photons were scattered back, allowing for the measurement of the interference in the electromagnetic field that resulted.
The improvement to this technique was they used solid-immersion lens technology. This is a lens that has a much higher numerical aperture than normal lenses as it is filled with a material that has a large refractive index. The technique used to measure the resonance fluorescence in this system was originally designed to locate individual molecules within substances.
The book first examines the applicability of the two-level model for atoms to real atoms, then explores semiclassical radiation theory, and derives the optical Bloch equations. It then examines Rabi inversion, optical nutation, free-induction decay, coherent optical transient effects, light amplification, superradiance, and photon echoes in solids and gases. Before the publication of this book, much of the material discussed was widely scattered in other books and research journals. This comprehensive treatment brings it together in one convenient resource. The style of writing is clear and informal and the emphasis throughout is always on the physics of the processes taking place. There are numerous helpful illustrations, excellent introductions to each chapter, and lists of references for further reading. Their coverage of the subject is remarkably complete.
We use a quantum-electrodynamical, many-body treatment to show mirrorless optical bistability in terms of the spatial properties of coherent dipole-dipole interactions among interacting two-level atoms. The general theory is applied to two special cases: 1 a thin sample of two-level atoms, with a width smaller than a resonance wavelength and 2 a long sample of two-level atoms, with dimensions very large relative to a resonance wavelength. While for the thin sample we are able to use a mean-field approximation with validity, for the long sample we are compelled to take into account retardation and propagation. In both cases bistability is found to be related to a renormalization of the frequency or relaxation rate that is inversion dependent. For the long sample the frequency renormalization is significant for high atomic densities and for large oscillator strengths.
Statistical Methods in Quantum Optics 1 pp Cite as. The damped harmonic oscillator provides our elementary description for the electromagnetic field in a lossy cavity. The damped two-level atom will provide our elementary description for the matter with which this field interacts. In an atomic vapor, loss of energy from an excited atom may take place via spontaneous emission or inelastic collisions. Elastic collisions can also play an important damping role; although, of course, they do not carry away energy; elastic collisions interrupt the phase of induced electronic oscillations and in this way damp the atomic polarization.
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Resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom. Typically the photon contained electromagnetic field is applied to the two-level atom through the use of a monochromatic laser. A two-level atom is a specific type of two-state system in which the atom can be found in the two possible states.
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